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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the complete $ {\rm GL}(n,{\bf C})$-decomposition of the stable cohomology of $ {\rm gl}\sb n(A)$

Author: Phil Hanlon
Journal: Trans. Amer. Math. Soc. 308 (1988), 209-225
MSC: Primary 17B55; Secondary 17B56
MathSciNet review: 946439
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Abstract: Let $ A$ be a graded, associative $ {\mathbf{C}}$-algebra. For each $ n$ let $ g{l_n}(A)$ denote the Lie algebra of $ n \times n$ matrices with entries from $ A$.

In this paper we extend the Loday-Quillen theorem to nontrivial isotypic components of $ GL(n,\,{\mathbf{C}})$ acting on the Lie algebra cohomology of $ g{l_n}(A)$. For $ \alpha $ and $ \beta $ partitions of some nonnegative integer $ m$ let $ {[\alpha ,\,\beta ]_n} \in {{\mathbf{Z}}^n}$ denote the maximal $ GL(n,\,{\mathbf{C}})$-weight given by

$\displaystyle {[\alpha ,\,\beta ]_n} = \sum\limits_i {{\alpha _i}{e_i}} - \sum\limits_j {{\beta _j}{e_{n + 1 - j}}.} $

We show that the $ {[\alpha ,\,\beta ]_n}$-isotypic component of the Lie algebra cohomology of $ g{l_n}(A)$ stabilizes when $ n \to \infty $ and is equal to

$\displaystyle HR{C^{\ast}}(A) \otimes ({\tilde H^{\ast}}{(A;\,{\mathbf{C}})^{ \otimes m}} \otimes {S^\alpha } \otimes {S^\beta }){s_m}$

where $ {\tilde H^{\ast}}(A;\,{\mathbf{C}})$ is the reduced Hochschild cohomology of $ A$ with trivial coefficients, where $ HR{C^{\ast}}(A)$ is the graded exterior algebra generated by the cyclic cohomology of $ A$, where $ {S^\alpha }$ and $ {S^\beta }$ are the irreducible $ {S_m}$-modules indexed by $ \alpha $ and $ \beta $ and where the action of $ {S_m}$ on $ \tilde H{(A;\,{\mathbf{C}})^{ \otimes m}}$ is the exterior action.

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Article copyright: © Copyright 1988 American Mathematical Society